jie-fang zhang- a class of coherent structures and interaction behavior in multidimensions

8/3/2019 Jie-Fang Zhang- A class of coherent structures and interaction behavior in Multidimensions

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A class of coherent structures and interaction

behavior in Multidimensions

Jie-Fang Zhang 1,2,3

1 Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua 321004, P.R.China

2 Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11,3TU,UK

3 Shanghai Institute of Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China

Abstract

We solve the (2+1)-dimensional Davey-Stewartson (DS) equation, a multidimensional

analog of the nonlinear Schrodinger equation. A rather general solution for the variables

separation with two arbitrary functions is first obtained by applying a special Backlund

transformation and introducing the seed solutions. And then some new special types of

two-dimensional coherent structures are obtained. These structures exhibit interesting novel

features not found in one-dimensional solitons.

Key words: coherent structures, variable separation, Davey-Stewartson(DS) equation

PACC: 03.40.Kf, 03.65.Ge

1 Introduction

We consider the Davey-Stewartson(DS)[1] system of equations:

iqt +1

2(qxx + qyy) (x + |q|2)q = 0, xx yy + 2|q|2x = 0, (1)

which is the shallow-water limit of the Benney-Rokes equation[2], where q is the amplitude of a

surface wave packet while characterizes the mean motion generated by this surface wave(One

assumes a small-amplitude, nearly monochromatic,nearly one-dimensional wave train with dom-

inant surface tension[3]). Equations (1) provides a two-dimensional generalization of the cele-

brated nonlinear Schrodinger equation. Furthermore, it arises generically in both physics and

mathematics.Indeed, it has been shown that a very large class of nonlinear dispersive equa-

tions in 2+1 (two spatial and one temporal) dimensions reduce to the (2+1)-dimensional DS

equation in appropriate but generic asymptotic considerations[4]. Physical applications include

water waves, Plasma physics, and nonlinear optics[5]. Fokas and Santini[6]have solved an initial-

boundary value problem for DS system by using the inverse scattering transform (IST) method

Corresponding author. E-mail:[email protected] address.

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and obtained a kind of two-dimensional coherent structures and found that the coherent struc-

tures of Eqs.(1) exhibit interesting novel features not found in one-dimensional solitons. In this

paper, we consider further Eqs.(1). A rather general solution of the variable separation with

two arbitrary functions is first obtained by applying a special Backlund transformation and

introducing the seed solutions. And then some new special types of two-dimensional coherentstructures are constructed. These structures exhibit also interesting novel features not found in

one-dimensional solitons.

2 Exact variable separation solution of the (2+1)-dimensional

DS system

It is convenient to introduce characteristic coordinates = x + y, = x

y, and U1

12|q|2, U2 12 |q|2, then the second equation(2) can be integrated and Eqs.(1)and (2) reduce

to

iqt + (q + q) + (U1 + U2)q = 0, (2)

U1 =1

2

|q|2d + 01, (3)

U2 =1

2

|q|2d + 02, (4)

where 01(, t) = U1(, , t), 02(, t) = U2(, , t).To find soliton solutions of an equation, we can use different kinds of methods. One of

the powerful methods is the variable separation approach, which was recently presented and

successfully applied in some (2+1)-dimensional models[7-11]. Now we would use this method

to investigate the (2+1)-dimensional DS equation. To solve the system (2), we first take the

following Backlund transformation

q =g

f+ q0, U1 = (ln f) + U10, U2 = (ln f) + U20, (5)

which can be obtained from the standard Painleve truncated expansion, where f is a real, g is

complex, and (q0, U10, U20) is an arbitrary seed solution. Substituting (5) directly into system

(2)-(4) and integrating Eqs.(3)and (4) once to the argument and respectively, yields its

bilinear form:

(D2 + D2 + iDt)g f + q0(D2 + D2)f f + 2f g(U10 + U20)

+2f2q0(U10 + U20) + f2(q0 + q0) + if

2q0t = 0, (6)

DDf f + 2(gg + f gq0 + f gq0 + f2q0q0 + 2f2(1 U10)) = 0, (7)

DDf f + 2(gg + f gq0 + f gq0 + f2q0q0 + 2f2(1 U20)) = 0, (8)

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where 1 , 1 are integration symbols to the argument and , and D, D, Dt are defined as

Dm Dn D

kt f f =lim=,=,t=t (

)m(

)n(

t

t)k (9)

which is the usual bilinear operator introduced first by Hirota[12].

To discuss further, we take the seed solution (q0, U10, U20) as

q0 = 0, U10 = u0(, t), U20 = v0(, t) (10)

then(6),(7)and (8) can be simplified to

(D2 + D2 + iDt)g f + 2f g(u0 + v0) = 0, (11)

DDf f + 2gg = 0. (12)

To find some interesting solutions of equations (11) and (12), we can use the variable separation

ansatz

f = a1u + a2v + a3uv, g = u1v1 exp(ir + is), (13)

where a1, a2, a3 are arbitrary constants and u u(, t), v v(, t), u1 u1(, t), v1 v1(, t), r r(, t), s s(, t) are all real functions of the indicated variables. SubstitutingEq.(13) into Eqs.(11) and (12) and separating the real and imaginary parts of the resulting

equations,we have

u21v21 2a1a2uv = 0, (14)

(a1u + a2v + a3uv)(v1u1 + u1v1 u1v1(2rt + 2st 2u0 2v0 + r2

+ s

2

))

+v1(a1 + a3v)(u1u 2u1u) + u1(a2 + a3u)(v1v 2v1v) = 0, (15)

(a1u + a2v + a3uv)(v1(2ru1 + 2u1t + u1r) + u1(2sv1 + 2v1t + v1s))

2u1v1(vt + 2sv)(a2 + a3u) 2u1v1(ut + 2ru)(a1 + a3v) = 0. (16)

Because the functions u0, u , u1 and r are only functions of{, t} and the functions v0, v, v1and s are only functions of {, t}, the equation system (14)-(16) can be solved by the followingvariable separated equations:

u1 = 12a1a2c10 u, (17)v1 = 2

c0v, (

21 =

22 = 1), (18)

ut + ru = c1(a2 + a3u)2 + c2(a2 + a3u) + a1a2c3, (19)

vt + sv = c3(a1 + a3v)2 c2(a1 + a3v) a1a2c1, (20)

4(2st + s2 2v0)v2 2vv + v2 + c4v2 = 0, (21)

4(2rt + r2 2u0)u2 2uu + u2 c4u2 = 0. (22)

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In Eqs.(17)-(22), c0, c1, c2, c3, c4 are all arbitrary functions of t. From Eqs.(17) and (18),

we know that the real conditions of u and v require

a1a2c10 u 0, (23)

c0v 0. (24)Although it is not an easy task to obtain general solutions of Eqs.(19)-(22) for any fixed u0

and v0, we can treat the problem in an alternative way. Because u0 and v0 are arbitrary seed

solutions, we can view u and v as arbitrary functions of {, t} and {, t} with the conditions(23), (24) respectively. The functions r and s can be expressed by u and v simply by integrations

from Eqs.(19) and (20). Then the seed solutions u0 and v0 can be fixed by Eqs.(21), (22). Finally,

substituting Eq.(13) with (17)-(22) into Eq.(8), we get a quite general solution of the (2+1)-

dimensional system (5)-(7)

q =12

a1a2uv exp(ir + is)

a1u + a2v + a3uv , (25)

U1 = u0 + (a1u + a3vu

a1u + a2v + a3uv (a1u + a3vu)

2

(a1u + a2v + a3uv)2), (26)

U2 = v0 + (a2v + a3uv

a1u + a2v + a3uv (a2v + a3uv)

2

(a1u + a2v + a3uv)2), (27)

with two arbitrary u and v under the conditions (24) and (25) and u0, v0 are determined by

Eqs.(22) and (23). Especially, for the module square of the field reads

=

|q

|2 =

a1a2uv(a1u + a2v + a3uv)2

(28)

=a1a2UV

2(A1 cosh1

2(U + V + C1) + A2 cosh

1

2(U V + C2))2

, (29)

where

u = b1 + eU, v = b2 + e

V , (30)

and

A1 =

a3(a1b1 + a2b2 + a3b1b2), A2 =

(a1 + a3b2)(a2 + a3b1), (31)

C1 = lna3

a1b1 + a2b2 + a3b1b2, C2 = ln

a1 + a3b2

a2 + a3b1, (32)

for b1 and b2 being arbitrary constants. U and V are also arbitrary functions of{, t} and {, t}respectively under the conditions

a1a2UV 0. (33)

Because u(, t) and v(, t) are arbitrary the functions of indicated arguments, Eq.(28) (or

Eq.(29))reveals the quite abundant soliton structures. From Eqs.(28) and (29), it is easy to

know that for arbitrary u and v with the boundary conditions

u

|

B1, u

|+

B2, v

|

B3, v

|+

B4, (34)

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where B1, B2, B3 and B4 are arbitrary constants which may be infinities.

We known that,in addition to the continuous localized excitations in (1+1)-dimensional non-

linear systems, some type of significant weak solutions like the peakon[13] and compacton[14]and

multi-valued localized solution like loop soliton[15]. The so-called peakon solution (u = c exp( |x ct |))which is called a weak solution of the celebrated (1+1)-dimensional Camassa-Holmequation

ut + 2kux uxxt + 3uux = 2uxuxx + uuxxx, (35)

was firstly given by Camassa and Holm[13]. While the so called (1+1)-dimensional compacton

solutions which describes the typical (1+1)-dimensional soliton solutions with finite wavelength

when the nonlinear dispersion effects are included was firstly given by Rosenau and Hyman[14].

It has found that peakon and compacton may have many interesting properties and possible

physical applications[16]. Moreover, in natural world, there exist very complicated folded phe-

nomena such as the folded protein[17]folded brain and skin surface and many many other kinds

of folded biology system[18]. The bubbles on (or under) a fluid surface may be thought to be the

simplest folded waves. Further, various kinds of ocean waves are really folded waves also. These

phenomena have been applied in some physical fields like the string interaction with external

field, quantum field theory and particle physics[19]. Recently, the higher dimensional peakon

solution and compacton solution and foldon solution, which are new types of soliton if the inter-

action between the folded solitary waves is completely elastic, have also been investigated and

obtained in some (2+1)-dimensional models[20-22]. Here we focus our attention on giving the

three kinds of interesting coherent structures from the expression(28) for the (2+1)-dimensional

DS system.

3 Compacton solutions and their interaction behavior

Because of the entrance of arbitrary functions in expression(26), we can easily find some types of

multiple compacton solutions by selecting the arbitrary functions appropriately. For instance,if

we fixed the functions u and v as

u =a0

a1+

M

i=1

0 + it 0i 2kibi sin(ki( + it

0i)) + bi 0i

2ki< + it

0i +

2ki,

2bi + it > 0i +2ki

(36)

and

v =N

j=1

0 + jt 0j 2li ,cj sin(lj( + jt 0j)) + cj 0j 2lj < + jt 0j + 2lj ,2ci + jt > 0j +

2lj

(37)

in the expressions (36) and (37), bi, ki, cj , i, j , lj , 0i and 0j are all arbitrary constants, then

the solution (28) with (36)and (37) becomes a multi-compactons solution.

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From(36) and (37), we can see that the piecewise functions u and v of the compacton solutions

are once differential

u =M

i=1

0 + it 0i 2kibiki cos(ki( + it

0i)) + bi 0i

2ki

< + it

0i +

2ki

,

0 + it > 0i +2ki

(38)

and

v =N

j=1

0 + jt 0j 2lj ,cj lj cos(lj( + jt 0j)) + cj 0j 2lj < + jt 0j + 2lj .0 + jt > 0j +

2lj

(39)

If selecting M = 2, N = 2, a0 = 20, a1 = a2 = 1, a3 =1

25, b1 = b2 = 1, c1 = 1, c2 = 1.5, k1 =

0.6, k2 = 0.4, 1 = 1, 2 = 3, 1 = 2 = 0, l1 = 1, l2 = 0.7, 01 = 02 = 0, 01 = 0, 02 = 5we can obtain a four-compactons excitation for the (2+1)-dimensional DS system. Fig.1(a)-(f)

shows the evolution behavior of interaction among four compactons. We see that the interaction

among four compactons exhibits a new phenomenon. Their interaction is non-elastic and do not

completely exchange their shapes each other.

4 Peakon solutions and their interaction behavior

Similarly, considering the arbitrariness of the functions u and v in expression (28), we can

construct the peakons of the (2+1)-dimensional DS system by selecting appropriate functions.

For instance, when u and v are taken the following simple form

u =a0a1

+Mi=1

di exp(mi it + 0i), mi it + 0i 0,di exp(mi + it 0i), mi it + 0i > 0,

(40)

v =N

j=1

ej exp(nj jt + y0j), nj jt + y0j 0,ej exp(nj + jt y0j) + 2, nj jt + y0j > 0,

(41)

where di, mi, ej , nj , i, j , 0 and 0j are all arbitrary constants, then the solution (28) with (40)

and(41)becomes a multi-peakon solution.

If we selecting M = 2, N = 2, a0 = 200, a1 = a2 = 1, a3 =1

200, d1 = d2 = 1, m1 = 0.5, m2 =

1, 1 = 1, 2 = 2e1 = e2 = 1, n1 = 1, n2 = 1, 1 = 1, = 2, 01 = 4, 02 = 4, 01 = 4, 02 =4, we can obtain a four-peakons excitation for the (2+1)-dimensional DS system. Fig.2(a)-(f)shows the evolution behavior of interaction among four peakons at different times. We can find

that the interaction of four-peakons is not completely elastic but four peakons may completely

exchange their shapes.

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5 Dromion solutions and their interaction behavior

In order to compare with the results given in Ref.[6], we also discuss the corresponding dromion

solution for the (2+1)-dimensional DS system.If selecting

u =Mi=1

tanh(ki + it), v =Ni=1

exp(lj + jt), (42)

we can obtain another kind of the multi-dromions solution. Figure 3(a)-(f)shows the evolution

behavior of interaction among four dromions when M = 4, N = 1, a0 = 10, a1 = a2 = a3 =

1, k1 = 1, k2 = 0.5, k3 = 1, k4 = 1, 1 = 0.5, 2 = 1, 3 = 3, 4 = 5, l1 = 1 and 1 = 1 at different

times. From figure 3(a)-(f), we can find that the interaction of four dromions is not completely

elastic and four dromions do not completely exchange their shapes each other.

6 Foldon solutions and and their interaction behavior

In order to construct these kinds of interesting folded solitary waves and/or foldons for the

module square of the field q, we should introduce some suitable multi-valued functions. For

example,

v =Mi=1

Vi( + cit), = +Mi=1

Yi( + it), (43)

where 1 < 2 < < Mare all arbitrary constants and Vi, Yi, j are localized functionswith properties Vi() = 0, Yi() = consts. From Eq.(43), one knows that may be a

multi-valued function in some suitable regions of by selecting the functions Yi appropriately.Therefore, the function v, which is obviously an interaction solution of M localized excitations

since the property | , may be a multi-valued function of in these areas though it isa single valued functions of .Actually, most of the known multi-loop solutions are the special

situations of Eq.(43). Similarly, we also treat the function u(, t) in this way

u =N

j=1

Uj( + jt), = +N

j=1

Xj( + jt), (44)

where 1 < 2 < < Nare all arbitrary constants andUj , Xj , j are all localized functions

with properties Uj() = 0, Xj() = consts. Now we further discuss the properties ofthe interaction among the folded solitary waves. If we select u and v to be some appropriate

multi-valued functions, then we can see that the interactions among the folded solitary waves

are completely elastic. For example, when set

u = sec h2(), = 2 tanh(), (45)

v =4

5sec h2() +

1

2sec h2( t/4),

= 32

(tanh() + tanh( t/4)), (46)

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and a1 = a2 = a3 = 1, then we can derived some interesting foldons depicted in Figs.(2a), (2b)

and (2c) at different times (a) t = 18, (b)t = 8, (c)t = 11.8,(d) t = 14,(e)t = 18 respectively.From figures (4a) and (4f), one can find the interaction between the two foldons is completely

elastic since the velocity of one of the foldons has set to be zero and there are still phase shifts for

the two foldons. To see more carefully, one can easily find that the position located by the largestatic foldon is altered from about = 1.5 to = 1.5 and its shape is completely preservedafter interaction.

7 Summary and discussion

In summary, with help of Backlund transformation and the variable separation procedures,

the (2+1)-dimensional DS system has also solved as Fokas and Santini by use of the inverse

scattering transform. By selecting arbitrary functions appropriately, three kinds of new coherent

structures(peakon, compacton and foldon) and a kind of known coherent structures (dromion)

have been constructed. The interactions among peakons and compactons exhibit interesting

novel features not found in one-dimensional solitons. The interaction among four foldons is

completely elastic. The interaction among four compactons or four dromions is non-elastic and

do not completely exchanged their shapes each other. While the interaction of four travelling

peakons or four dromions is not non-elastic but four peakons may completely exchange their

shapes.

Acknowledgement: The author is in debt to thank the useful discussions with Professor

Lou Senyue. Appreciation is also given to Prof. Roger Grimshaw, Dr. Rod Halburd and Dr.Zhiming Lu for their kind help. This work is supported by the Pao Yu-Kong and Pao Zhao-Long

Scholarship for Chinese Students Studying Abroad and the the Natural Science Foundation of

China Granted No.10272072.

References

[1] Davey A and Stewartson K 1974 Proc. Roy. Soc. London A 338 101

[2] Benney D J and Roskes G J 1969 Stud. Appl.Math. 48 377

[3] Ablowitz M J and Segur H 1981 Solitons and Inverse Scattering Transform(SIAM)

[4] Calogero F and Eckhaus W 1987 Inverse Problems 3L27

[5] Nishinari K and Abe K and J Satsuma 1993 J. Phys. Soc. Jpn.62 2021; Schultz C L Ablowitz

and BarYaacov D 1987 Phys. Rev. Lett, 59 2825; Pang G D Pu F C and Zhao B H 1990

Phys.Rev. Lett. 65 3227

[6] Fokas A S and Santini P M 1989 Phys. Rev Lett. 63 1329; Physica D 1990 4499

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[7] Lou S Y 2000 Phys. Lett. A 277 94;2000 Phys. Lett. A 277 94; 2001 Physica Scripta64 1;

Lou S Y, Chen C L and Tang X Y 2002 J. Math. Phys 43 4078;Lou S Y and Ruan H Y

2001 J. Phys. A: Math. Gen.34 305

[8] Ruan H Y and Chen Y X 2001 Acta Phys.Sin.50 586(in Chinese)

[9] Lou S Y, Lin J and Tang X Y 2001 Eur. Phys. J. B22 473; Lou S Y, Tang X Y and Chen

C L 2002 Chin. Phys. Lett. 19 769

[10] Zhang J F 2002 Commun. Ther. Phys. 37 277; 2002 Acta Phys.Sin. 51 705 (in Chinese)

[11] Zheng C L and Zhang J F 2002 Chin. Phys. Lett. 19 1399

[12] Hirota R 1971 Phys.Rev.Lett.27 1192

[13] Camassa R and Holm D D 1993 Phys. Rev. Lett. 71 1661

[14] Rosenau P and Hyman J M 1993 Phys. Rev. Lett.70 564

[15] Trewick S C, Henshaw T F,Hansinger R P, Lindahl T and Sedgwick B 2002 Nature 419

174 ;

Lockless S W,Ranganathan R 1999 Science 268 295 ;

Lindagard P A andBohr H 1996 Phys.Rev.Lett. 77 779

[16] Qian T and M Tiao 2001 Chao Soliton and Fractals 12 1347;

Kraenkel R A Senthilvelan M and Zenchunk 2000 Phys. Lett. A 273 183;

Cooper F Hyman J M and Khare A 2001 Phys. Re. E 64 026608 ;

Chertock A and Levy D 2001 J.Comput. Phys.171 708;

Manna M A 2001 Physica D 149 231

[17] Goodman M B,Ernstrom G G,Chelur D S, Hagan R O, Yao C A and Chalfie M 2002 Nature

4151039;

Maclnnis B L, Campenot R B 2002 Science 295 1536

[18] Vakhnenko V O 1992 J.Phys.A:Math.Gen.25 4181;

Vakhnenko V O and Parks E J 1998 Nonlinearity 11 1457;

Morrison A J,Parks E J and Vakhnenko V O 1999 ibid. 121427

[19] Kakuhata H and Konno K 1999 J.Phys.Soc.Jpn.68 757;

Matsutani S 1995 Mod.Phys.Lett.A,10 717(); 2002 J.Geom.Phys.43 146;

Schleif M and Wunschm R 1998 Eur.J.Phys.A 1 171;

Schleif M, Wunsch R and Meissner B T 1998 Int.J.Mod.Phys.E,7 121

[20] Tang X Y, Lou S Y and Zhang Y 2002 Phys. Rew. E66 46601

[21] Lou S Y 2002 J. Phys. A 35 10619

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[22] Tang X Y and Lou S Y 2002 arXiv. nlin. SI/0210009. 4 1; 2003 J. Math. Phys. in press

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Fig.1a

50

510

2

0 2

46

8

0

0.0005

0.001

Fig.1b

42

0

2 46

2

02

46

8

0

0.0005

0.001

Fig.1c

42

02

4

2

02

46

8

0

0.0005

0.001

0.0015

Fig.1d

86

42

0

2 4

2

02

46

8

0

0.0005

0.001

Fig.1e

108

64

20

24

2

02

46

8

0

0.0005

Figure 1: Evolution plot of a four-compactons solution determined by (28) with (36) and (37) at (a)

t = 3, (b) t = 0.9, (c) t = 0.02, (d) t = 1, (e) t = 1.8.

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Fig.2a

108

64

2

024

108

64

20 2

4

0

1e05

2e05

Fig.2b

86

42

02

86

42

02

01e052e053e05

Fig.2c

54

32

10

12

54

32

10

1 2

0

4e05

8e05

Fig.2d

6

4

2

0

2

6

4

2

0

2

0

1e05

2e05

Fig.2e

86

42

02

46

8

86

42

02

46

8

0

1e05

2e05

Figure 2: Evolution plot of a four-peakons solution determined by (28) with (40) and (41) at (a) t = 5,(b) t = 3, (c) t = 2, (d) t = 1.4, (e) t = 0.5.

12


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Fig.3a

010

203040

50

4

8

12

16

0

0.001

0.002

0.003

0.004

Fig.3b

50

510

1520

4

0

4

8

0

0.001

0.002

0.003

0.004

Fig.3c

4321

0123

64

20

24

6

0

0.002

0.004

0.006

0.008

Fig.3d

86

42

02

8

4

0

4

0

0.001

0.002

0.003

0.004

0.005

Fig.3e

2520

1510

50

8

4

0

4

0

0.001

0.002

0.003

0.004

Figure 3: Evolution plot of a four-dromions solution determined by (28) with (44) at (a) t = 10, (b)t = 3, (c) t = 0, (d) t = 1, (e)t = 4.

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Fig.4a

2

1

0

1

2

3.5

32.5

21.51

0

0.0005

0.001

0.0015

0.002

Fig.4b

2

1

0

1

2

0

0.51

1.52

0

0.0004

0.0008

0.0012

0.0016

Fig.4c

2

1

0

1

2

1

1.21.4

1.61.8

0

0.0005

0.001

0.0015

Fig.4d

2

1

0

1

2

1.21.62

2.4

0

0.0005

0.001

0.0015

0.002

Fig.4e

2

1

0

1

2

1

1.52

2.53

3.5

0

0.0005

0.001

0.0015

0.002

Figure 4: Evolution plots of two foldons for the module square of the field q given by Eq.(28) with

conditions (45), (46) and a1 = a2 = 1, a3 =1

20at different times (a) t = 18, (b) t = 8, (c) t = 11.8, (d)

t = 14, (e) t = 18

jie-fang zhang- a class of coherent structures and interaction behavior in multidimensions

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